In a periodically driven system, the central object for the description
of topological properties is the Floquet operator, which is the evolution
operator of the system over a complete driving period \(T\) . Taking
the square root of such a propagator for the purpose of generating
its topological descendants is, however, a highly nontrivial task.
This can be seen by writing the Floquet operator as \(U={\cal T}e^{-\frac{i}{\hbar }\int _{0}^{T}H(t)dt}=e^{-i\frac{T}{\hbar }H_{{\rm eff}}}\) ,
where \({\cal T}\)  is the time-ordering operator, \(H(t)=H(t+T)\)  is
the time-periodic Hamiltonian of the system, and \(H_{{\rm eff}}\) 
is the Floquet effective Hamiltonian obtained by formally working
out the time-ordered product in \({\cal T}e^{-\frac{i}{\hbar }\int _{0}^{T}H(t)dt}\) .
We may now take the square-root of \(U\)  to obtain a Floquet system
naively as \(\sqrt{U}=e^{-i\frac{T}{\hbar }\frac{H_{{\rm eff}}}{2}}\) .
However, such a trial of generating square-root Floquet topological
phases tends out to be problematic and useless. First, the exact form
of \(H_{{\rm eff}}\)  can be rather complicated (usually including driving-induced
long-range coupling terms), not physically obtainable, or even insufficient
to describe Floquet phases with no static counterparts such as those with anomalous edge modes [1]}, [2]}, [3]}.
Second, there are no transparent ways to find \(H_{{\rm eff}}\)  from
\(H(t)\) , i.e., the parameters in \(H_{{\rm eff}}\)  are usually nonlinear
combinations of physical parameters in \(H(t)\) , such that simply reducing
the parameters of \(H(t)\)  by half could not yield \(H_{{\rm eff}}/2\) .
Even obtained, the \(H_{{\rm eff}}\)  and \(H_{{\rm eff}}/2\)  describe
essentially the same physical system up to a global constant, and
no new physics are expected to emerge following such a halving process. Therefore,
the straightforward operation, \(\sqrt{U}=e^{-i\frac{T}{\hbar }\frac{H_{{\rm eff}}}{2}}\) ,
does not generate a desired square-root of the parent system described
by \(U\) .
